% ============================================================================= % BibTeX Bibliography for: % "The Riemann Hypothesis via Discrete Spectral Theory: % Zeros of the Zeta Function as Eigenvalues of a % Self-Adjoint Lattice Operator" % Author: Rafael D. De Paz % ============================================================================= % --- The Riemann Hypothesis --- @article{Riemann1859, author = {Riemann, Bernhard}, title = {\"Uber die {A}nzahl der {P}rimzahlen unter einer gegebenen {G}r\"o\ss{}e}, journal = {Monatsberichte der K\"oniglich Preu\ss{}ischen Akademie der Wissenschaften zu Berlin}, year = {1859}, pages = {671--680} } @misc{ClayRH, author = {{Clay Mathematics Institute}}, title = {The {R}iemann {H}ypothesis}, year = {2000}, url = {https://www.claymath.org/millennium/riemann-hypothesis/}, note = {Millennium Prize Problem, official problem statement by E.~Bombieri} } @article{Bombieri2000, author = {Bombieri, Enrico}, title = {Problems of the {M}illennium: The {R}iemann {H}ypothesis}, journal = {Clay Mathematics Institute Millennium Prize Problems}, year = {2000}, url = {https://www.claymath.org/millennium/riemann-hypothesis/} } % --- Analytic Number Theory Background --- @book{Titchmarsh1986, author = {Titchmarsh, E. C.}, title = {The Theory of the {R}iemann Zeta-Function}, publisher = {Oxford University Press}, year = {1986}, edition = {2nd}, note = {Revised by D. R. Heath-Brown} } @book{Davenport2000, author = {Davenport, Harold}, title = {Multiplicative Number Theory}, publisher = {Springer-Verlag}, year = {2000}, edition = {3rd}, doi = {10.1007/978-1-4757-5927-3} } @book{IwaniecKowalski2004, author = {Iwaniec, Henryk and Kowalski, Emmanuel}, title = {Analytic Number Theory}, publisher = {American Mathematical Society}, year = {2004}, isbn = {978-0-8218-3633-0} } % --- The Hilbert-Polya Conjecture and Spectral Theory --- @unpublished{HilbertPolya, author = {Hilbert, David and P\'olya, George}, title = {Unpublished correspondence suggesting a spectral interpretation of the {R}iemann zeros}, year = {1914}, note = {Communicated to A.~Odlyzko, see \url{https://www.dtc.umn.edu/~odlyzko/polya/index.html}} } @article{BerryKeating1999, author = {Berry, Michael V. and Keating, Jonathan P.}, title = {The {R}iemann Zeros and Eigenvalue Asymptotics}, journal = {SIAM Review}, volume = {41}, number = {2}, pages = {236--266}, year = {1999}, doi = {10.1137/S0036144598347497} } @article{BerryKeating1999b, author = {Berry, Michael V. and Keating, Jonathan P.}, title = {{$H = xp$} and the {R}iemann Zeros}, journal = {Supersymmetry and Trace Formulae: Chaos and Disorder}, editor = {Lerner, I. V. and Keating, J. P. and Khmelnitskii, D. E.}, publisher = {Kluwer Academic}, pages = {355--367}, year = {1999} } @article{Connes1999, author = {Connes, Alain}, title = {Trace formula in noncommutative geometry and the zeros of the {R}iemann zeta function}, journal = {Selecta Mathematica}, volume = {5}, number = {1}, pages = {29--106}, year = {1999}, doi = {10.1007/s000290050042} } @article{ConnesCons2021, author = {Connes, Alain and Consani, Caterina}, title = {The {R}iemann-{R}och strategy, complex lift of the scaling site}, journal = {Advances in Mathematics}, volume = {391}, pages = {107984}, year = {2021}, doi = {10.1016/j.aim.2021.107984} } % --- Random Matrix Theory --- @article{Montgomery1973, author = {Montgomery, Hugh L.}, title = {The pair correlation of zeros of the zeta function}, booktitle = {Analytic Number Theory}, series = {Proceedings of Symposia in Pure Mathematics}, volume = {24}, pages = {181--193}, publisher = {American Mathematical Society}, year = {1973}, doi = {10.1090/pspum/024/9944} } @article{Odlyzko1987, author = {Odlyzko, Andrew M.}, title = {On the distribution of spacings between zeros of the zeta function}, journal = {Mathematics of Computation}, volume = {48}, number = {177}, pages = {273--308}, year = {1987}, doi = {10.1090/S0025-5718-1987-0866115-0} } @article{KeatingSnaith2000, author = {Keating, Jonathan P. and Snaith, Nina C.}, title = {Random Matrix Theory and $\zeta(1/2 + it)$}, journal = {Communications in Mathematical Physics}, volume = {214}, number = {1}, pages = {57--89}, year = {2000}, doi = {10.1007/s002200000261} } % --- Discrete / Spectral Methods --- @article{AtsuyamaKatsura1979, author = {Akatsuka, Hirotaka}, title = {Zeta functions of finite graphs}, journal = {Journal of the Mathematical Society of Japan}, volume = {55}, pages = {559--574}, year = {2003} } @article{Ihara1966, author = {Ihara, Yasutaka}, title = {On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields}, journal = {Journal of the Mathematical Society of Japan}, volume = {18}, number = {3}, pages = {219--235}, year = {1966}, doi = {10.2969/jmsj/01830219} } @article{HaShieh2013, author = {Hashimoto, Ki-ichiro}, title = {Zeta functions of finite graphs and representations of $p$-adic groups}, journal = {Advanced Studies in Pure Mathematics}, volume = {15}, pages = {211--280}, year = {1989} } @article{Sunada1986, author = {Sunada, Toshikazu}, title = {$L$-functions in geometry and some applications}, journal = {Curvature and Topology of Riemannian Manifolds}, series = {Lecture Notes in Mathematics}, volume = {1201}, pages = {266--284}, year = {1986}, publisher = {Springer} } @article{Terras1999, author = {Terras, Audrey}, title = {A survey of discrete trace formulas}, journal = {Emerging Applications of Number Theory}, series = {IMA Volumes in Mathematics and its Applications}, volume = {109}, publisher = {Springer}, year = {1999}, pages = {643--681} } % --- Self-Adjoint Operators --- @book{ReedSimon1978, author = {Reed, Michael and Simon, Barry}, title = {Methods of Modern Mathematical Physics. {IV}: {A}nalysis of Operators}, publisher = {Academic Press}, year = {1978} } @book{Schmudgen2012, author = {Schm\"udgen, Konrad}, title = {Unbounded Self-Adjoint Operators on {H}ilbert Space}, publisher = {Springer}, year = {2012}, doi = {10.1007/978-94-007-4753-1} } % --- Selberg Trace Formula --- @article{Selberg1956, author = {Selberg, Atle}, title = {Harmonic analysis and discontinuous groups in weakly symmetric {R}iemannian spaces with applications to {D}irichlet series}, journal = {Journal of the Indian Mathematical Society}, volume = {20}, pages = {47--87}, year = {1956} } @book{Hejhal1976, author = {Hejhal, Dennis A.}, title = {The {S}elberg Trace Formula for {$\mathrm{PSL}(2, \mathbb{R})$}}, publisher = {Springer-Verlag}, series = {Lecture Notes in Mathematics}, volume = {548}, year = {1976} } % --- Weil Explicit Formula --- @article{Weil1952, author = {Weil, Andr\'e}, title = {Sur les ``formules explicites'' de la th\'eorie des nombres premiers}, journal = {Communications du S\'eminaire Math\'ematique de l'Universit\'e de Lund}, pages = {252--265}, year = {1952}, note = {Dedicated to M. Riesz} } % --- Quantum Chaos --- @article{GutzwillerTrace, author = {Gutzwiller, Martin C.}, title = {Periodic Orbits and Classical Quantization Conditions}, journal = {Journal of Mathematical Physics}, volume = {12}, number = {3}, pages = {343--358}, year = {1971}, doi = {10.1063/1.1665596} } @book{StockmannBook, author = {St\"ockmann, Hans-J\"urgen}, title = {Quantum Chaos: An Introduction}, publisher = {Cambridge University Press}, year = {1999}, doi = {10.1017/CBO9780511524622} } % --- Planck Scale --- @article{Planck1899, author = {Planck, Max}, title = {\"Uber irreversible {S}trahlungsvorga\"nge}, journal = {Sitzungsberichte der K\"oniglich Preu\ss{}ischen Akademie der Wissenschaften zu Berlin}, year = {1899}, pages = {440--480} } @article{montgomery1973pair, title={The pair correlation of zeros of the zeta function}, author={Montgomery, Hugh L}, journal={Analytic number theory}, pages={181--193}, year={1973}, publisher={American Mathematical Soc.} } @article{odlyzko1987distribution, title={On the distribution of spacings between zeros of the zeta function}, author={Odlyzko, Andrew M}, journal={Mathematics of Computation}, volume={48}, number={177}, pages={273--308}, year={1987} }